A mathematician works with basic shapes and theorems.A scientist suggests theorems and computes the probability of something being correct or not, and also where and when it is valid.An engineer likes to build large things that shoot out of large things just for the sake of creating something huge.A bioengineer takes the engineering mentality a step further with biological "creations".

Basically each is a step further leading to bioengineering.

The chemistry note is just an extra joke that has no relation to the actual comic. Zach probably added it so people don't ask why he did not mention it. It doesn't fit in the ladder to begin with.